X = 36 degrees
Call the apex point of the triangle A
Call the angle bisector AD, where D is the point where the bisector intersects the base of the triangle.
Let the vertexes counterclockwise from A be labeled B and C.
Then, in triangle ADC, AD = DC. And in any triangle having equal sides, the angles opposite those sides are equal, too. Therefore, < DCA (angle "X") = < DAC (one of the angles formed by the bisector).
And in triangle ABD, AB = AD, so the measure of angles ABD and ADB are equal.
And by the exterior angle theorem, angle ADB = DAC + DCA. Therefore, ABD = BAD + DCA.
And, because of bisection, angle BAD = DAC. And angle BAC equals their sum.
Therefore, ABD + BAC + DCA = 180.
Therefore, by substitution, BAD + DCA + BAD + DAC + DCA = 180 or, simplifying, 2BAD + 2DCA + DAC = 180.
But DCA = DAC., And DAC = BAD.
Therefore, DCA = DAC = BAC = BAD.
And substituting again......2DCA + 2DCA + DCA = 180
Therefore, 5DCA=180
Therefore DCA = 36 = X
Let's check that this is true. In triangle ABD....ABD = 72 angle ADB =72 and BAD = 36....these sum to 180
And, in triangle ADC, angle ADC is supplemental to angle ADB (i.e., 108) and DAC = 36 and DCA = 36...so these sum to 180, as well.
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